Question

A randomized controlled trial was designed to compare the effectiveness of splinting versus surgery in the treatment of carpal tunnel syndrome. Results are given in the table below (based on data from “Splinting vs. Surgery in the Treatment of Carpal Tunnel Syndrome,” by Gerritsen et al., Journal of the American Medical Association,Vol. 288,

No. 10). The results are based on evaluations made one year after the treatment. Using a 0.01 significance level, test the claim that success is independent of the type of treatment. What do the results suggest about treating carpal tunnel syndrome?

	Successful Treatment	Unsuccessful Treatment
Splint Treatment	60	23
Surgery Treatment	67	6

Short answer

Success is dependent on the type of treatment and the type of treatment (splint or surgery) would determine the success of treatment in the carpal tunnel syndrome.

Step-by-step solution

Given information

The data forthe success of treatment and type of treatment is provided.

The level of significance is 0.01.

Compute the expected frequencies and check the requirements

Theexpected frequency is computed as,

\(E = \frac{{\left( {row\;total} \right)\left( {column\;total} \right)}}{{\left( {grand\;total} \right)}}\)

The table for the observed values and the totals of row and column is represented as,

	Successful Treatment	Unsuccessful Treatment	Row total
Splint Treatment	60	23	83
Surgery Treatment	67	6	73
Column total	127	29	156

Theexpected frequency tableis represented as,

	Successful Treatment	Unsuccessful Treatment
Splint Treatment	67.571	15.429
Surgery Treatment	59.429	13.571

Here, all expected values are greater than 5.

Assuming the subjects are gathered from random selection and assigned randomly to the groups, the requirements of chi-square test is satisfied.

State the null and alternate hypothesis

The claim for independence of type of treatment and the success of treatment is tested using the following hypotheses:

\({H_0}:\)Success is independent of the type of treatment

\({H_1}:\)Success is dependent of the type of treatment.

Compute the test statistic

The value of the test statisticis computed as,

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {60 - 67.571} \right)}^2}}}{{67.571}} + \frac{{{{\left( {23 - 15.429} \right)}^2}}}{{15.429}} + ... + \frac{{{{\left( {6 - 13.571} \right)}^2}}}{{13.571}}\\ = 9.750\end{aligned}\]

Therefore, the value of the test statistic is 9.750.

Compute the degrees of freedom

The degrees of freedomare computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {2 - 1} \right)\\ = 1\end{aligned}\)

Therefore, the degrees of freedom are 1.

Compute the critical value

From chi-square distribution table, the critical value for row corresponding to 1 degrees of freedom and at 0.01 level of significance is 6.635.

Also, the p-value is obtained as 0.002.

State the decision

Since the critical (6.635) is less than the value of test statistic (9.750). In this case, the null hypothesis is rejected.

Therefore, the decision is to reject the null hypothesis.

State the conclusion

There is not enough evidencein favour of the claim that success is independent of the type of treatment.

It suggests that the choice of treatment would be essential to determine the success of the treatment in the carpal tunnel syndrome.